Axisymmetric Irrotational Flow in Spherical Coordinates

In an irrotational flow pattern, we can automatically satisfy the constraint $\nabla\times {\bf v}={\bf0}$ by writing

$${\bf v} = -\nabla\phi.$$ (7.11)

Suppose, however, that, in addition to being irrotational, the flow pattern is also incompressible: that is, $\nabla\cdot{\bf v}=0$ . In this case, Equation (7.11) yields

$$\nabla^{\,2}\phi = 0.$$ (7.12)

In spherical coordinates, assuming that the flow pattern is axisymmetric, so that $\phi=\phi(r,\theta)$ , the previous equation leads to (see Section C.4)

$$\frac{1}{r^{\,2}}\,\frac{\partial}{\partial r}\left(r^{\,2}\,\fra... ...{\partial\theta}\left(\sin\theta\,\frac{\partial\phi}{\partial\theta}\right)=0.$$ (7.13)

Let us search for a separable solution of Equation (7.13) of the form

$$\phi(r,\theta)= R(r)\,{\mit\Theta}(\theta).$$ (7.14)

It is easily seen that

$$\frac{1}{R}\,\frac{d}{dr}\left(r^{\,2}\,\frac{dR}{dr}\right) = -\... ...heta}\,\frac{d}{d\theta}\left(\sin\theta\,\frac{d{\mit\Theta}}{d\theta}\right),$$ (7.15)

which can only be satisfied provided

$$\frac{d}{dr}\left(r^{\,2}\,\frac{dR}{dr}\right) -l\,(l+1)\,R = 0,$$ (7.16)

$$\frac{d}{d\mu}\left[(1-\mu^{\,2})\,\frac{d{\mit\Theta}}{d\mu}\right]+l\,(l+1)\,{\mit\Theta} = 0,$$ (7.17)

where $\mu=\cos\theta$ , and $l\,(l+1)$ is a constant. The solutions to Equation (7.17) that are well behaved for $\mu$ in the range $-1$ to $+1$ are known as the Legendre polynomials, and are denoted the $P_l(\mu$ ), where $l$ is a non-negative integer (Jackson 1962). (If $l$ is non-integer then the solutions are singular at $\mu=\pm 1.$ ) In fact,

$$P_l(\mu)= \frac{(-1)^{\,l}}{2^{\,l}\,l!}\,\frac{d^{\,l}}{d\mu^{\,l}}\left(1-\mu^{\,2}\right)^l.$$ (7.18)

Hence,

$$P_0(\mu) = 1,$$ (7.19)

$$P_1(\mu) =\mu,$$ (7.20)

$$P_2(\mu) = \frac{1}{2}\left(3\,\mu^{\,2}-1\right),$$ (7.21)

$$P_3(\mu) = \frac{1}{2}\left(5\,\mu^{\,3}-3\,\mu\right),$$ (7.22)

et cetera. The general solution of Equation (7.16) is a linear combination of $r^{\,l}$ and $r^{-(l+1)}$ factors. Thus, the general axisymmetric solution of Equation (7.12) is written

$$\phi(r,\theta) = \sum_{l=0,\infty} \left[\alpha_l\,r^{\,l} + \beta_l\,r^{-(l+1)}\right]P_l(\cos\theta),$$ (7.23)

where the $\alpha_l$ and $\beta_l$ are arbitrary coefficients. It follows from Equations (7.4) that the corresponding expression for the Stokes stream function is

$$\psi(r,\mu)= \beta_0\,\mu+\sum_{l=1,\infty}\left(\frac{\alpha_l}{... ...1}-\frac{\beta_l}{l}\,r^{\,-l}\right)\left(1-\mu^{\,2}\right)\frac{dP_l}{d\mu},$$ (7.24)

where $\mu=\cos\theta$ .